Homogeneous function: Difference between revisions

Revision as of 14:46, 1 September 2022

Revision as of 14:48, 1 September 2022

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{{Short description|Function with a multiplicative scaling behaviour}}

{{More footnotes|date=July 2018}}

{{for|homogeneous linear maps|Graded vector space#Homomorphisms}}

In [[mathematics]], a '''homogeneous function''' is a [[function of several variables]] such that, if all its arguments are multiplied by a [[scalar (mathematics)|scalar]], then its value is multiplied by some power of this scalar, called the '''degree of homogeneity''', or simply the ''degree''; that is, if {{mvar|k}} is an integer, a function {{mvar|f}} of {{mvar|n}} variables is homogeneous of degree {{mvar|k}} if

:<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>

for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math>

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===Homogeneous polynomials===

{{main article|Homogeneous polynomial}}

[[Monomials]] in <math>n</math> variables define homogeneous functions <math>f : \mathbb{F}^n \to \mathbb{F}.</math> For example,

<math display="block">f(x, y, z) = x^5 y^2 z^3 \,</math>

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{{or section|date=December 2021}}

Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex</math>). If <math>S</math> is a set of scalars, such as <math>\Z</math>, <math>[0, \infty)</math>, or <math>\R</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if

<math display="inline">f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in S</math>.

For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R</math>.